Chapter 12: Q10P (page 518)
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You probably did Prob. 12.4 from the point of view of an observer on the ground. Now do it from the point of view of the police car, the outlaws, and the bullet. That is, fill in the gaps in the following table:
| Speed of Relative to | Ground | Police | Outlaws | Bullet | Do they escape? |
| Ground | 0 | role="math" localid="1654061605668" | |||
| Police | |||||
| Outlaws | |||||
| Bullet |
In classical mechanics, Newtonโs law can be written in the more familiar form . The relativistic equation, , cannot be so simply expressed. Show, rather, that
where is the ordinary acceleration.
Consider a particle in hyperbolic motion,
(a) Find the proper time role="math" localid="1654682576730" as a function of , assuming the clocks are set so that when . [Hint: Integrate Eq. 12.37.]
(b) Find x and v (ordinary velocity) as functions of .
(c) Find (proper velocity) as a function of .
You may have noticed that the four-dimensional gradient operator functions like a covariant 4-vectorโin fact, it is often written , for short. For instance, the continuity equation, , has the form of an invariant product of two vectors. The corresponding contravariant gradient would be . Prove that is a (contravariant) 4-vector, if is a scalar function, by working out its transformation law, using the chain rule.
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